Disjoint paths in hypercubes with prescribed origins and lengths

Abstract

Given (i) any k vertices u 1, u 2, …, u k (1≤k<n) in the n-cube Q n , where (u 1, u 2), (u 3, u 4), …, (u 2m−1, u 2m ) (m≤⌊ k\\2 ⌋) are edges of the same dimension, (ii) any k positive integers a 1, a 2, …, a k such that a 1, a 2, …, a 2m are odd and a 2m+1, …, a k are even, with a 1+a 2+···+a k =2 n , and (iii) k subsets W 1, W 2, …, W k of V(Q n ) with |W i |≤n−k and if a i =1, then u i ¬∈W i , for 1≤i≤k, we show that there exist k vertex-disjoint paths P (1), P (2), …, P (k) in Q n where P (i) contains a i vertices, its origin is u i , and its terminus is in V(Q n )/ W i , for 1≤i≤k. We also prove a similar result which extends two well-known results of Havel, [I. Havel On hamilton circuits and spanning trees of hypercubes, Časopis pro Pĕstování Matematiky, 109 (1984), pp. 135–152.] and Nebeský, [L. Nebeský Embedding m-quasistars into n-cubes, Czech. Math. J. 38 (1988), pp. 705–712].